Introduction to Octonion and Other Non-Associative Algebras in Physics

Description

In this book, the author applies non-associative algebras to physics. Okubo covers topics ranging from algebras of observables in quantum mechanics and angular momentum and octonions to division algebra, triple-linear products and YangSHBaxter equations. He also discusses the non-associative gauge theoretic reformulation of Einstein's general relativity theory. Much of the material found in this volume is not available in other works. The book will therefore be of great interest to graduate students and research scientists in physics and mathematics.

This book was written by one of the luminaries in the field of mathematical physics. The book begins with an epic paragraph: "The saying that God is the mathematician, so that, even with meager experimental support, a mathematically beautiful theory will ultimately have a greater chance of being correct, has been attributed to Dirac. The octonion algebra may surely be called a beautiful mathematical entity. Nevertheless, it has never been systematically utilized in physics in any fundamental fashion, although some attempts have been made toward this goal. However, it is still possible that nonassociative algebras (other than Lie algebras) may play some essential future role in the ultimate theory, yet to be discovered.''
The main text of this book consists of nine chapters in which octonion algebras and its variants and symmetries are omnipresent with reference to various physical implications and applications. Other non-associative algebras or triple systems, for example, alternative, flexible, Lie-admissible and Malʹtsev algebras, are also treated to generalize or reformulate such topics as Yang-Mills gauge theory, the Heisenberg approach to quantum mechanics, the Schrödinger equation, the Yang-Baxter equation, Chern-Simons theory and Einstein's general relativity theory. This is probably the first account which extensively discusses the interplay between various classes of non-associative algebras, other than Lie algebras, and contemporary subjects in theoretical physics. Other physical relevances, that are not covered in the main text, are cited with suitable references in the Remarks (there are 59 numbered remarks). The interested reader may be tempted to further explore some of these relevances, for example, relationships of superalgebras, Jordan pairs and triple systems with both quantum and classical mechanics, Nambu dynamics, quantum propositional calculus and nonlinear Schrödinger equations.
The first five chapters concern basic structures of some well-known non-associative algebras. In Chapter 1, the multiplication of the octonion (Cayley) algebra is recast using totally anti-symmetric tensors in dimensions 3 and 4, and the role of octonions in physics is emphasized in a similar fashion to those of quaternions and Pauli spin matrices. A brief introduction is given in Chapter 2 to Jordan, Lie, flexible, alternative, Lie-admissible and Jordan-admissible algebras. In Chapter 3, the structure of Hurwitz (unital composition) algebras is discussed in more detail with a proof for the dimensionality using an argument based on Clifford algebras. As an application, the octonion algebra is used to obtain the instanton solution of the Yang-Mills field equation.
Chapter 4 concerns two important classes of non-unital composition algebras, called para-Hurwitz and pseudo-octonion algebras. The latter algebras were first introduced by the author in 1978 to generalize the Heisenberg approach in quantum mechanics [Hadronic J. 1 (1978), no. 4, 1250–1278; MR0510100], and its forms are now called Okubo algebras [see A. Elduque and H. C. Myung, Comm. Algebra 19 (1991), no. 4, 1197–1227; MR1102335; Comm. Algebra 21 (1993), no. 7, 2481–2505; MR1218509]. It has recently been shown that the flexible, more generally, third-power composition algebras comprise Hurwitz, para-Hurwitz and Okubo algebras (over an arbitrary field of characteristic Malʹtsev 2,3) (for details, see papers by the reviewer [Non-unital composition algebras, Seoul Nat. Univ., Seoul, 1994; MR1286256] and Elduque and J. M. Pérez [Manuscripta Math. 84 (1994), no. 1, 73–87; MR1283328]). The first two classes of algebras above are Malʹtsev-admissible while any Okubo algebra is Lie-admissible [see H. C. Myung, Malʹtsev-admissible algebras, Progr. Math., 64, Birkhäuser Boston, Boston, MA, 1986; MR0885089].
Let C(p,q) denote the real Clifford algebra defined by Dirac's matrix relations. The main result in Chapter 5 is the classification of real matrix representations of C(p,q), using the theorem of Frobenius on real associative division algebras. It follows from this result that the Majorana-Weyl spinor is possible for C(p,q) if and only if p≡q(mod8). The Majorana-Weyl spinor is an important tool in superstring theory. A nonassociative Dirac algebra of dimension 12(N2+3N+4) with N generators is constructed, which is neither flexible nor power-associative. This algebra is Lie-admissible and its associated Lie algebra contains so(N+2) as a subalgebra. An expanded version of Chapter 5 appears in earlier papers by the author [J. Math. Phys. 32 (1991), no. 7, 1657–1668; MR1112690; J. Math. Phys. 32 (1991), no. 7, 1669–1673; MR1112691; Progr. Theoret. Phys. Suppl. No. 86 (1986), 287–296; MR0868852].
For a positive integer J (spin number), the irreducible su(2)-module V with a basis of the 2J+1 angular momentum state vectors is made into a non-associative algebra V (a Clebsch-Gordan algebra) using the Clebsch-Gordan decomposition. When J=1 or 3, V yields the Lie algebra so(3) or the 7-dimensional simple real (compact) Malʹtsev algebra. Adjoining a unit [resp. para-unit] to V, the quaternion [resp. para-quaternion] or the octonion [resp. para-octonion] algebra is obtained from this construction. More generally, Chapter 6 concerns the construction of various nonassociative algebras or triple products defined on an L-module V for a Lie algebra L such that L acts as derivations on V. This is done using the module homomorphisms of Young-tableau components of V⊗V, V⊗V⊗V into V (or F, the base field). Particular cases are emphasized for L=G2 and its subalgebras su(3), su(2) and so(4), which again reproduce some of the aforementioned algebras. For topics related to Chapter 6, see also the reviewer's book cited above.
In Chapter 7, Jordan and flexible Lie-admissible algebras are discussed as algebras of physical observables in the viewpoint of generalizing quantum mechanics. For the latter algebras, the Heisenberg approach is emphasized because they satisfy the Leibnitz rule for the adjoint maps (see also the reviewer's book above) and assure the self-consistency of quantization. If a Jordan algebra of observables is used for the Heisenberg approach, then all observables are time-independent, corresponding to a kind of Schrödinger representation. A Lie-admissible (not flexible) algebra is constructed from observables in this view with Schrödinger wave functions.
In Chapter 8, the author returns to the method employed in Chapter 6 to construct various triple products on an L-module with a particular emphasis in alternative triples and in those interplaying closely with octonion algebras. The (quaternionic and octonionic) triple systems V considered satisfy certain identities (implying dimV=4 or 8) and provide solutions to the Yang-Baxter equation via a new solution method (for details, see two papers by the author [J. Math. Phys. 34 (1993), no. 7, 3273–3291; MR1224212; J. Math. Phys. 34 (1993), no. 7, 3292–3315; MR1224213]). As another application, Nambu's approach to modifying the Heisenberg equation of motion is discussed using triple systems, in particular, alternative and Lie triple systems. A more interesting case for Nambu dynamics arises when the triple product {x,y,z} is totally antisymmetric, since two Hamiltonians are conserved in this case. Some physical implications of Jordan pairs and Lie and Jordan triple super-systems are noted in this chapter.
In Chapter 9, a nonassociative gauge theory is developed in a way that reproduces the standard Yang-Mills gauge theory when the given nonassociative algebra A is a Lie algebra. The underlying idea for this is to utilize AutA as the transformation group of the gauge transformation (notice that AutA is a Lie group with the derivation algebra DerA as its Lie aglebra). The author reformulates Einstein's general relativity theory using a nonassociative Chern-Simons theory. Chapter 10 concludes this book with a list of other applications and relevances of nonassociative algebras to physics, along with a series of references.
This book will remain an excellent source for applications of nonassociative algebras in physics for a long time.

Reviewed by Hyo C. Myung